Mixing
Evaluate $int_{0}^{2pi} lfloorcot^{-1}xrfloor,dx$ from $0$ to $2 pi$.
$begingroup$
Evaluate $int_{0}^{2pi} lfloorcot^{-1}xrfloor,dx$ from $0$ to $2 pi$. Where $lfloor{}cdot{}rfloor$ denotes the greatest integer function.
Here $x$ varies from $0$ to $2 pi$, so
$cot^{-1}0 > cot^{-1}x > cot^{-1}2 pi$
$infty > cot^{-1}x > cot^{-1}2 pi$
Now
$int_{cot1}^{ 2pi} lfloorcot^{-1}xrfloor,dx + int_{0}^{cot1}lfloorcot^{-1}xrfloor,dx$
The first part of integral is clearly zero, how to evaluate second part?
calculus
edited Jan 17 at 14:20
egreg
183k1486204
asked Jan 17 at 14:11
MathsaddictMathsaddict
3619
$endgroup$
add a comment |
$begingroup$
Evaluate $int_{0}^{2pi} lfloorcot^{-1}xrfloor,dx$ from $0$ to $2 pi$. Where $lfloor{}cdot{}rfloor$ denotes the greatest integer function.
Here $x$ varies from $0$ to $2 pi$, so
$cot^{-1}0 > cot^{-1}x > cot^{-1}2 pi$
$infty > cot^{-1}x > cot^{-1}2 pi$
Now
$int_{cot1}^{ 2pi} lfloorcot^{-1}xrfloor,dx + int_{0}^{cot1}lfloorcot^{-1}xrfloor,dx$
The first part of integral is clearly zero, how to evaluate second part?
calculus
edited Jan 17 at 14:20
egreg
183k1486204
asked Jan 17 at 14:11
MathsaddictMathsaddict
3619
$endgroup$
$begingroup$
@egreg you edited the question correctly.
$endgroup$
– Mathsaddict
Jan 17 at 14:19
$begingroup$
@Mathsaddict The symbol $lfloor{}cdot{}rfloor$ is more commonly used than $[{}cdot{}]$.
$endgroup$
– egreg
Jan 17 at 14:21
$begingroup$
math.stackexchange.com/questions/3076606/…
$endgroup$
– lab bhattacharjee
Jan 17 at 14:26
add a comment |
$begingroup$
Evaluate $int_{0}^{2pi} lfloorcot^{-1}xrfloor,dx$ from $0$ to $2 pi$. Where $lfloor{}cdot{}rfloor$ denotes the greatest integer function.
Here $x$ varies from $0$ to $2 pi$, so
$cot^{-1}0 > cot^{-1}x > cot^{-1}2 pi$
$infty > cot^{-1}x > cot^{-1}2 pi$
Now
$int_{cot1}^{ 2pi} lfloorcot^{-1}xrfloor,dx + int_{0}^{cot1}lfloorcot^{-1}xrfloor,dx$
The first part of integral is clearly zero, how to evaluate second part?
calculus
edited Jan 17 at 14:20
egreg
183k1486204
asked Jan 17 at 14:11
MathsaddictMathsaddict
3619
$endgroup$
Evaluate $int_{0}^{2pi} lfloorcot^{-1}xrfloor,dx$ from $0$ to $2 pi$. Where $lfloor{}cdot{}rfloor$ denotes the greatest integer function.
Here $x$ varies from $0$ to $2 pi$, so
$cot^{-1}0 > cot^{-1}x > cot^{-1}2 pi$
$infty > cot^{-1}x > cot^{-1}2 pi$
Now
$int_{cot1}^{ 2pi} lfloorcot^{-1}xrfloor,dx + int_{0}^{cot1}lfloorcot^{-1}xrfloor,dx$
The first part of integral is clearly zero, how to evaluate second part?
calculus
calculus
edited Jan 17 at 14:20
egreg
183k1486204
asked Jan 17 at 14:11
MathsaddictMathsaddict
3619
edited Jan 17 at 14:20
egreg
183k1486204
asked Jan 17 at 14:11
MathsaddictMathsaddict
3619
edited Jan 17 at 14:20
egreg
183k1486204
edited Jan 17 at 14:20
egreg
183k1486204
edited Jan 17 at 14:20
egreg
183k1486204
183k1486204
asked Jan 17 at 14:11
MathsaddictMathsaddict
3619
asked Jan 17 at 14:11
MathsaddictMathsaddict
3619
asked Jan 17 at 14:11
MathsaddictMathsaddict
3619
3619
$begingroup$
@egreg you edited the question correctly.
$endgroup$
– Mathsaddict
Jan 17 at 14:19
$begingroup$
@Mathsaddict The symbol $lfloor{}cdot{}rfloor$ is more commonly used than $[{}cdot{}]$.
$endgroup$
– egreg
Jan 17 at 14:21
$begingroup$
math.stackexchange.com/questions/3076606/…
$endgroup$
– lab bhattacharjee
Jan 17 at 14:26
add a comment |
$begingroup$
@egreg you edited the question correctly.
$endgroup$
– Mathsaddict
Jan 17 at 14:19
$begingroup$
@Mathsaddict The symbol $lfloor{}cdot{}rfloor$ is more commonly used than $[{}cdot{}]$.
$endgroup$
– egreg
Jan 17 at 14:21
$begingroup$
math.stackexchange.com/questions/3076606/…
$endgroup$
– lab bhattacharjee
Jan 17 at 14:26
$begingroup$
@egreg you edited the question correctly.
$endgroup$
– Mathsaddict
Jan 17 at 14:19
$begingroup$
@egreg you edited the question correctly.
$endgroup$
– Mathsaddict
Jan 17 at 14:19
$begingroup$
@Mathsaddict The symbol $lfloor{}cdot{}rfloor$ is more commonly used than $[{}cdot{}]$.
$endgroup$
– egreg
Jan 17 at 14:21
$begingroup$
@Mathsaddict The symbol $lfloor{}cdot{}rfloor$ is more commonly used than $[{}cdot{}]$.
$endgroup$
– egreg
Jan 17 at 14:21
$begingroup$
math.stackexchange.com/questions/3076606/…
$endgroup$
– lab bhattacharjee
Jan 17 at 14:26
$begingroup$
math.stackexchange.com/questions/3076606/…
$endgroup$
– lab bhattacharjee
Jan 17 at 14:26
add a comment |
1 Answer
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$begingroup$
$$0le xlecot1$$
$$dfracpi2gecot^{-1}xge1impliesleftlfloorcot^{-1}xrightrfloor=1$$
answered Jan 17 at 14:18
lab bhattacharjeelab bhattacharjee
226k15157275
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
$$0le xlecot1$$
$$dfracpi2gecot^{-1}xge1impliesleftlfloorcot^{-1}xrightrfloor=1$$
answered Jan 17 at 14:18
lab bhattacharjeelab bhattacharjee
226k15157275
$endgroup$
add a comment |
$begingroup$
$$0le xlecot1$$
$$dfracpi2gecot^{-1}xge1impliesleftlfloorcot^{-1}xrightrfloor=1$$
answered Jan 17 at 14:18
lab bhattacharjeelab bhattacharjee
226k15157275
$endgroup$
add a comment |
$begingroup$
$$0le xlecot1$$
$$dfracpi2gecot^{-1}xge1impliesleftlfloorcot^{-1}xrightrfloor=1$$
answered Jan 17 at 14:18
lab bhattacharjeelab bhattacharjee
226k15157275
$endgroup$
$$0le xlecot1$$
$$dfracpi2gecot^{-1}xge1impliesleftlfloorcot^{-1}xrightrfloor=1$$
answered Jan 17 at 14:18
lab bhattacharjeelab bhattacharjee
226k15157275
answered Jan 17 at 14:18
lab bhattacharjeelab bhattacharjee
226k15157275
answered Jan 17 at 14:18
lab bhattacharjeelab bhattacharjee
226k15157275
answered Jan 17 at 14:18
lab bhattacharjeelab bhattacharjee
226k15157275
226k15157275
add a comment |
add a comment |
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$begingroup$
@egreg you edited the question correctly.
$endgroup$
– Mathsaddict
Jan 17 at 14:19
$begingroup$
@Mathsaddict The symbol $lfloor{}cdot{}rfloor$ is more commonly used than $[{}cdot{}]$.
$endgroup$
– egreg
Jan 17 at 14:21
$begingroup$
math.stackexchange.com/questions/3076606/…
$endgroup$
– lab bhattacharjee
Jan 17 at 14:26